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In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers
$p< q$
do the equalities
$T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$
,
$k=p, q$
, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if
$p$
is odd and
$q$
is even, and negative otherwise. The case
$(p,q)=(1,2)$
closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on
$C^*$
-algebras is also provided.
We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys.186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys.116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.
We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra
$\mathfrak {g}$
satisfies a polynomial identity if and only if the nilpotent radical
$\mathfrak {n}$
of
$\mathfrak {g}$
is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on
$\mathfrak {n}$
.
We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.
Let
$C_{\||.\||}$
be an ideal of compact operators with symmetric norm
$\||.\||$
. In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on
$[0,\infty)$
and S and T are bounded operators in
$\mathbb{B}(\mathscr{H}\;\,)$
such that
${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$
, then
In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.
The classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.
In this note, we mainly study operator-theoretic properties on the Besov space
$B_{1}$
on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.
Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.
We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.
Given a holomorphic self-map
$\varphi $
of
$\mathbb {D}$
(the open unit disc in
$\mathbb {C}$
), the composition operator
$C_{\varphi } f = f \circ \varphi $
,
$f \in H^2(\mathbb {\mathbb {D}})$
, defines a bounded linear operator on the Hardy space
$H^2(\mathbb {\mathbb {D}})$
. The model spaces are the backward shift-invariant closed subspaces of
$H^2(\mathbb {\mathbb {D}})$
, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.
Let u and
$\varphi $
be two analytic functions on the unit disk D such that
$\varphi (D) \subset D$
. A weighted composition operator
$uC_{\varphi }$
induced by u and
$\varphi $
is defined on
$A^2_{\alpha }$
, the weighted Bergman space of D, by
$uC_{\varphi }f := u \cdot f \circ \varphi $
for every
$f \in A^2_{\alpha }$
. We obtain sufficient conditions for the compactness of
$uC_{\varphi }$
in terms of function-theoretic properties of u and
$\varphi $
. We also characterize when
$uC_{\varphi }$
on
$A^2_{\alpha }$
is Hilbert–Schmidt. In particular, the characterization is independent of
$\alpha $
when
$\varphi $
is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on
$A^2_{\alpha }$
.
Li et al. [A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267(12) (2014), 4753-4774] proved a spectral radius type formula for the approximation numbers of composition operators on analytic Hilbert spaces with radial weights and on $H^{p}$ spaces, $p\geq 1$, involving Green capacity. We prove that their formula holds for a wide class of Banach spaces of analytic functions and weights.
We study the isomorphic structure of
$(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$
and prove that these spaces are complementably homogeneous. We also show that for any operator T from
$(\sum {\ell }_{q})_{c_{0}}$
into
${\ell }_{q}$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
that is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
and the restriction of T on X has small norm. If T is a bounded linear operator on
$(\sum {\ell }_{q})_{c_{0}}$
which is
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular, then for any
$\epsilon>0$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
which is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
with
$\|T|_{X}\|<\epsilon $
. As an application, we show that the set of all
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular operators on
$(\sum {\ell }_{q})_{c_{0}}$
forms the unique maximal ideal of
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$
.
The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.
We investigate unbounded, linear operators arising from a finite sum of composition operators on Fock space. Real symmetry and complex symmetry of these operators are characterised.
The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in
$(0,\infty)$
. The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted
$L_1$
-space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of
${\mathrm {H}}^2$
turns out to be false.
Let
$\mathcal {X}$
be a Banach space over the complex field
$\mathbb {C}$
and
$\mathcal {B(X)}$
be the algebra of all bounded linear operators on
$\mathcal {X}$
. Let
$\mathcal {N}$
be a nontrivial nest on
$\mathcal {X}$
,
$\text {Alg}\mathcal {N}$
be the nest algebra associated with
$\mathcal {N}$
, and
$L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
be a linear mapping. Suppose that
$p_n(x_1,x_2,\ldots ,x_n)$
is an
$(n-1)\,$
th commutator defined by n indeterminates
$x_1, x_2, \ldots , x_n$
. It is shown that L satisfies the rule
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
if and only if there exist a linear derivation
$D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
and a linear mapping
$H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$
vanishing on each
$(n-1)\,$
th commutator
$p_n(A_1,A_2,\ldots , A_n)$
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
such that
$L(A)=D(A)+H(A)$
for all
$A\in \text {Alg}\mathcal {N}$
. We also propose some related topics for future research.